Density of values of linear maps on quadratic surfaces

In this paper we investigate the density properties of the set of values of a linear map at integer points on a quadratic surface. In particular we show that this set is dense in the range of the linear map subject to certain algebraic conditions on the linear map and the quadratic form that defines the surface. The proof uses Ratner's theorem on orbit closures of unipotent subgroups acting on homogeneous spaces.